1
Journal of Integer Sequences, Vol. 14 (2011),
Article 11.6.3
2
3
47
6
23 11
A Note on a Problem of Motzkin Regarding
Density of Integral Sets with Missing
Differences
Ram Krishna Pandey1
Department of Mathematics
Indian Institute of Technology
Patliputra Colony, Patna – 800013
India
ram@iitp.ac.in
Amitabha Tripathi2
Department of Mathematics
Indian Institute of Technology
Hauz Khas, New Delhi – 110016
India
atripath@maths.iitd.ac.in
Abstract
For a given set M of positive integers, a problem of Motzkin asks to determine the
maximal density µ(M ) among sets of nonnegative integers in which no two elements
differ by an element of M . The problem is completely settled when |M | ≤ 2, and some
partial results are known for several families of M when |M | ≥ 3. In 1985 Rabinowitz
& Proulx provided a lower bound for µ({a, b, a + b}) and conjectured that their bound
was sharp. Liu & Zhu proved this conjecture in 2004. For each n ≥ 1, we determine
κ({a, b, n(a + b)}), which is a lower bound for µ({a, b, n(a + b)}), and conjecture this
to be the exact value of µ({a, b, n(a + b)}).
1
Introduction
For x ∈ R and a set S of nonnegative integers, let S(x) denote the number of elements n ∈ S
such that n ≤ x. The upper and lower densities of S, denoted by δ(S) and δ(S) respectively,
1
are given by
δ(S) := lim sup
x→∞
S(x)
,
x
δ(S) := lim inf
x→∞
S(x)
.
x
If δ(S) = δ(S), we denote the common value by δ(S), and say that S has density δ(S).
Given a set of positive integers M , S is said to be an M -set if a ∈ S, b ∈ S imply a − b ∈
/ M.
Motzkin [15] asked to determine µ(M ) given by
µ(M ) := sup δ(S)
S
where S varies over all M -sets. Cantor & Gordon [3] showed the existence of µ(M ) for any
M , determined µ(M ) when |M | ≤ 2, and gave the following lower bound for µ(M ):
µ(M ) ≥ κ(M ) :=
1
min |cmi |m ,
gcd(c,m)=1 m i
sup
(1)
where mi are the elements of M , and |x|m denotes the absolute value of the absolutely least
remainder of x modulo m. We use the following equivalent form for the lower bound for
µ(M ), due to Haralambis [11]:
κ(M ) = max
m=mi +mj
1≤k≤ m
2
1
min |kmi |m ,
m mi ∈M
(2)
where mi , mj represent distinct elements of M . The following useful upper bound for µ(M )
is due to Haralambis [11]:
µ(M ) ≤ α
(3)
provided there exists a positive integer k such that S(k) ≤ (k + 1)α for every M -set S with
0 ∈ S and for some α ∈ [0, 1]. The problem of Motzkin has a rich and diverse history but
little progress towards the general problem has been made so far. Exact results for µ(M )
have been few, and computation of µ(M ) has only been completely possible when |M | ≤ 2.
Cantor & Gordon [3] showed that
1
µ({m}) = ,
2
µ({m1 , m2 }) =
⌊(m1 + m2 )/2⌋
.
m1 + m2
There have, however, been a number of results that give the exact value or bounds for µ(M )
in other cases; see [13] for an exhaustive bibliography.
Connections with coloring problems in graph theory have been found useful in solving the
Motzkin problem. One such connection, introduced by Hale [10] and shown to be equivalent
to the Motzkin problem by Griggs & Liu [9], is the T -coloring problem. Another connection
with colorings of graphs involves the fractional chromatic number of distance graphs.
The lower bound for µ(M ), denoted by κ(M ) in (1), is itself at the heart of a longstanding
conjecture. The Lonely Runner Conjecture (LRC) stated independently by Wills [17] in the
context of diophantine approximations and by Cusick [7] while studying view obstructions
problems in n-dimensional geometry, was actually given this apt name by Bienia et al [1].
2
Chen [6] characterized 3-sets M for which 1/κ(M ) is an integer and also obtained general
bounds.
We consider the problem of determining µ(M ) for the family M = {a, b, n(a + b)}, n ≥ 1.
By a result of Cantor & Gordon [3], we know that µ(kM ) = µ(M ). Thus, it is no loss of
generality to assume that gcd(a, b) = 1, and that a < b. We determine the value of κ(M ),
which is the lower bound for µ(M ) and conjecturally equal to it. This extends a result of
Liu & Zhu [[13], Theorem 5.1], wherein they determined µ(M ) in the case M = {a, b, a + b}.
Rabinowitz & Proulx [16] provided a lower bound for µ(M ) in this case and conjectured that
their bound was sharp. An extensive list of work related to the Motzkin problem may be
found in [13].
2
Main Result
For the set M = {a, b, a + b}, Rabinowitz & Proulx [16] conjectured the exact value of
µ(M ) in 1985, and Liu & Zhu [13] proved this conjecture in 2004. We determine κ(M ) for
M = {a, b, n(a + b)} where n is a fixed positive integer. If the conjecture of Haralambis [11]
is true, then κ(M ) = µ(M ) in this case, thus providing a generalization of the result of Liu
& Zhu.
Theorem 1. Let M = {a, b, n(a + b)}, where a < b, gcd(a, b) = 1 and n ≥ 1. Then

n(a+b−λ) 
, if λ ≡ a + b (mod 2);

2 a+n(a+b)
κ(M ) = n(a+b+λ−1)

6 a + b (mod 2),
 2 b+n(a+b) , if λ ≡
where λ = 1 +
b−a
2n+1
.
Proof. We use (2) to compute κ(M ). The choice m = a+b trivially yields min |ax|m , |bx|m , |n(a+
b)x|m = 0 for each x. There are two choices remaining for m, and we determine κ(M ) by
comparing the two rational numbers corresponding to these cases. In each of the two cases
we need to compute κ(M ) with m = a + n(a + b) and m = b + n(a + b), and compare the
two. Note that the definition of λ implies
b − a < (2n + 1)λ ≤ b − a + 2n + 1.
Case I: λ ≡ a + b (mod 2)
Subcase (i): m = a + n(a + b) Choose x such that
(a + b)x ≡
a+b−λ
2
(mod m).
Then
−ax ≡ n(a + b)x ≡ n a+b−λ
≡
2
m−(a+nλ)
2
(mod m),
and
bx = (a + b)x − ax ≡
m+{b−(n+1)λ}
2
3
(mod m).
Therefore
min |ax|m , |bx|m , |n(a + b)x|m =
m−(a+nλ)
2
(4)
since b − (n + 1)λ < a + nλ if and only if (2n + 1)λ > b − a.
Write ℓ := a + nλ. We show that
min |ay|m , |by|m , |n(a + b)y|m ≤ m−(a+nλ)
= m2 − 2ℓ
2
for each y, 1 ≤ y ≤ m2 . Let I := m2 − 2ℓ , m2 + 2ℓ . We show that −ay mod m ∈ I and
by mod m ∈ I is simultaneously impossible for 1 ≤ y ≤ m2 . Suppose
(a + b)y ≡
a+b−λ
2
+ i (mod m),
with 1 ≤ i ≤ m − 1. Then
− ay ≡ n(a + b)y ≡ m2 − 2ℓ + ni (mod m),
+ (n + 1)i (mod m)
by ≡ (a + b)y − ay ≡ m2 − 2ℓ + a+b−λ
2
(5)
Thus −ay mod m ∈ I if and only if
km +
m
2
−
ℓ
2
<
m
2
− 2ℓ + ni < km +
m
2
+
ℓ
2
for some integer k, with 0 ≤ k ≤ n − 1. This is equivalent to
km
< i < km
+ nℓ ,
n
n
so that
k(a + b) + 1 ≤ i ≤ k(a + b) + a + λ − 1.
(6)
For k(a + b) + 1 ≤ i ≤ k(a + b) + a + λ − 1, we show that
km + m2 + 2ℓ < m2 − 2ℓ + a+b−λ
+ (n + 1) k(a + b) + 1
2
and
m
2
− 2ℓ +
a+b−λ
2
+ (n + 1) k(a + b) + a + λ − 1 < (k + 1)m +
m
2
− 2ℓ .
This will prove that
km +
m
2
+
ℓ
2
< by < (k + 1)m +
m
2
− 2ℓ ,
so that by mod m ∈
/ I , as claimed. Each of the above two inequalities is easy to prove. Using
the fact that (n + 1)(a + b) = m + b, each inequality can be shown to hold if (2n + 1)λ <
(b − a) + 2(n + 1), which is true by the definition of λ. This completes the subcase when
m = a + n(a + b).
Subcase (ii): (m = b + n(a + b)) The argument in this subcase is similar to the one in
subcase (i). We omit the calculation and state only the significant parts. Choose x such
that
(a + b)x ≡ a+b+λ
(mod m).
2
4
Then
−bx ≡ n(a + b)x ≡
m−(b−nλ)
2
(mod m),
and
ax = (a + b)x − bx ≡ − m−{a+(n+1)λ}
2
(mod m).
Therefore
min |ax|m , |bx|m , |n(a + b)x|m =
m−{(a+(n+1)λ}
2
(7)
since b − nλ < a + (n + 1)λ if and only if (2n + 1)λ > b − a.
As in subcase (i), we may show that
min |ay|m , |by|m , |n(a + b)y|m ≤
for each y, 1 ≤ y ≤
subcase (ii).
m
.
2
m−{a+(n+1)λ}
2
The argument is similar and we omit the proof. This completes
To compute κ(M ) in Case I, we need to compare the expressions in (4) and (7). If we let
m1 = a + n(a + b) and m2 = b + n(a + b), then a lengthy but easy computation shows that
m2 −{(a+(n+1)λ}
2m2
=
1
2
−
1 a+(n+1)λ
2 b+n(a+b)
<
1
2
−
1 a+nλ
2 a+n(a+b)
=
m1 −(a+nλ)
2m1
if and only if (2n + 1)λ > b − a. This completes the proof of Case I.
Case II: λ 6≡ a + b (mod 2)
Subcase (i): (m = a + n(a + b)) Choose x such that
(a + b)x ≡
a+b−λ+1
2
(mod m).
Then
−ax ≡ n(a + b)x ≡
m−{a+n(λ−1)}
2
(mod m),
and
bx = (a + b)x − ax ≡
m+{b−(n+1)(λ−1)}
2
(mod m).
Therefore
min |ax|m , |bx|m , |n(a + b)x|m =
m−{b−(n+1)(λ−1)}
2
since a + n(λ − 1) ≤ b − (n + 1)(λ − 1) if and only if (2n + 1)(λ − 1) ≤ b − a.
As in subcase (i) of Case I, we may show that
min |ay|m , |by|m , |n(a + b)y|m ≤
for each y, 1 ≤ y ≤
m
.
2
m−{b−(n+1)(λ−1)}
2
This completes subcase (i).
Subcase (ii): (m = b + n(a + b)) Choose x such that
(a + b)x ≡
a+b+λ−1
2
5
(mod m).
(8)
Then
−bx ≡ n(a + b)x ≡
m−{b−n(λ−1)}
2
(mod m),
and
ax = (a + b)x − bx ≡
m+{a+(n+1)(λ−1)}
2
(mod m).
Therefore
min |ax|m , |bx|m , |n(a + b)x|m =
m−{b−n(λ−1)}
2
(9)
since a + (n + 1)(λ − 1) ≤ b − n(λ − 1) if and only if (2n + 1)(λ − 1) ≤ b − a.
As in subcase (i) of Case I, we may show that
min |ay|m , |by|m , |n(a + b)y|m ≤
for each y, 1 ≤ y ≤
m
.
2
m−{b−n(λ−1)}
2
This completes subcase (ii).
To compute κ(M ) in Case II, we need to compare the expressions in (8) and (9). If we let
m1 = a + n(a + b) and m2 = b + n(a + b), then a lengthy but easy computation shows that
m1 −{b−(n+1)(λ−1)}
2m1
=
1
2
−
1 b−(n+1)(λ−1)
2 a+n(a+b)
≤
1
2
−
1 b−n(λ−1)
2 b+n(a+b)
=
m2 −{b−n(λ−1)}
2m2
if and only if (2n + 1)(λ − 1) ≤ b − a. This completes the proof of Case II, and of the
theorem.
Corollary 2. (Liu & Zhu [13])
Let M = {a, b, a + b}, where a < b and gcd(a, b) = 1. Then
1
if b ≡ a (mod 3);

3,
2a+b−1
κ(M ) = 3(2a+b) , if b ≡ a + 1 (mod 3);

 a+2b−1
, if b ≡ a + 2 (mod 3).
3(a+2b)
Proof. This is a direct consequence of Theorem 1. Set b − a = 3k + r, where 0 ≤ r ≤ 2.
Then λ = k + 1, so that we are in Case I when b ≡ a + 1 (mod 3) and Case II otherwise.
The calculation is routine, and the details are omitted.
Remark 3. Haralambis [11] conjectured that µ(M ) = κ(M ) when |M | = 3. If this is true,
Theorem 1 actually determines µ({a, b, n(a + b)}). Observe that, if a, b are of opposite parity
and if n ≥ (b − a)/2, Theorem 1 reduces to
κ({a, b, n(a + b)}) =
a+b−1
,
2(a + b) + (2a/n)
which is asymptotic to µ({a, b}). This may be an indication that the conjecture of Haralambis
may hold, at least for the special case M = {a, b, n(a + b)} when n is large enough, and
perhaps even for M = {a, b, c} when c is sufficiently large even if not of the form n(a + b).
6
3
Acknowledgement
The authors wish to thank the referee for his comments.
References
[1] W. Bienia, L. Goddyn, P. Gvozdjak, A. Sebö, and M. Tarsi, Flows, view obstructions
and the lonely runner, J. Combin. Theory Ser. B 72 (1998), 1–9.
[2] J. Barajas and O. Serra, The lonely runner with seven runners, Electron. J. Combin.
15 (2008), Research paper 48, 18 pages.
[3] D. G. Cantor and B. Gordon, Sequences of integers with missing differences, J. Combin.
Theory Ser. A 14 (1973), 281–287.
[4] G. J. Chang, L. Huang, and X. Zhu, The circular chromatic numbers and the fractional
chromatic numbers of distance graphs, European J. Combin. 19 (1998), 423–431.
[5] G. Chang, D. Liu, and X. Zhu, Distance graphs and T -colorings, J. Combin. Theory
Ser. B 75 (1999), 159–169.
[6] Y. G. Chen, On a conjecture about diophantine approximations III, J. Number Theory
37 (1991), 181–198.
[7] T. W. Cusick, View-obstruction problems in n-dimensional geometry, J. Combin. Theory Ser. A 16 (1974), 1–11.
[8] R. B. Eggleton, P. Erdős, and D. K. Skilton, Coloring the real line, J. Combin. Theory
Ser. B 39 (1985), 86–100.
[9] J. R. Griggs and D.-D. F. Liu, The channel assignment problem for mutually adjacent
sites, J. Combin. Theory Ser. A 68 (1994), 169–183.
[10] W. K. Hale, Frequency assignment: theory and applications, Proceedings of the IEEE
68 (1980), 310–320.
[11] N. M. Haralambis, Sets of integers with missing differences, J. Combin. Theory Ser. A
23 (1977), 22–33.
[12] A. Kemnitz and H. Kolberg, Coloring of integer distance graphs, Discrete Math. 191
(1998), 113–123.
[13] D.-D. F. Liu and X. Zhu, Fractional chromatic number for distance graphs with large
clique size, J. Graph Theory 47 (2004), 129–146.
[14] D. D.-F. Liu and X. Zhu, Fractional chromatic number of distance graphs generated by
two-interval sets, European J. Combin. 29 (7) (2008), 1733–1742.
7
[15] T. S. Motzkin, Unpublished problem collection.
[16] J. H. Rabinowitz and V. K. Proulx, An asymptotic approach to the channel assignment
problem, SIAM J. Alg. Disc. Methods 6 (1985), 507–518.
[17] J. M. Wills, Zwei Sätze uber inhomogene diophantische Approximation von Irrationalzahlen, Monatsh. Math. 71 (1967), 263–269.
2010 Mathematics Subject Classification: Primary 11B05.
Keywords: density, coloring.
Received January 6 2011; revised version received May 18 2011. Published in Journal of
Integer Sequences, June 1 2011.
Return to Journal of Integer Sequences home page.
8